We consider a class of nonlocal conservation laws with exponential kernel and prove that quantities involving the nonlocal term W:=𝟙(−∞,0](⋅)exp(⋅)∗ρ satisfy an Oleĭnik-type entropy condition. More precisely, under different sets of assumptions on the velocity function V, we prove that W satisfies a one-sided Lipschitz condition and that V′(W)W∂xW satisfies a one-sided bound, respectively. As a byproduct, we deduce that, as the exponential kernel is rescaled to converge to a Dirac delta distribution, the weak solution of the nonlocal problem converges to the unique entropy-admissible solution of the corresponding local conservation law, under the only assumption that the initial datum is essentially bounded and not necessarily of bounded variation.